Comments for MEDB 5502, Week 03

Topics to be covered

What you will learn
- Indicator variables for three or more categories
- Multiple factor analysis of variance
- Checking assumptions of analysis of variance
- Interactions in analysis of variance
- Interactions in analysis of covariance
- Interactions in multiple linear regression
- The general linear model
- To be determined

Review oneway analysis of variance

\(H_0:\ \mu_1=\mu_2=...=\mu_k\)
\(H_1:\ \mu_i \ne \mu_j\) for some i, j
- Reject \(H_0\) if F-ratio is large
Note: when k=2, use analysis of variance or t-test

Full moon data

Admission rates to mental health clinic before, during, and after full moon.
One year of data

Boxplot of full moon data

Descriptive statistics

Analysis of variance table

Tukey post hoc

Creating indicator variables

Running general linear model with all indicator variables

Analysis of variance with first and second indicators

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Parameter estimates, 1 of 3

11.458 - 13.417 = -1.959
10.917 - 13.417 = -2.5

Here are the parameter estimates. The intercept is the estimated average rate when the first and second indicator both equal zero. This is effectively the average rate for the third group (during a full moon).

The first slope is the estimated average difference in rates when you increase the first indicator by one unit and hold the second indicator constant. This is effectively the the average rate for the first group (after a full moon) minus the average rate for the third group (during a full moon). When the days switch from during a full moon to after a full moon, the estimated average admission rate decreases by almost two patients per day.

The second slope is the estimated average difference in rates when you increase the second indicator by one unit and hold the first indicator constant. This is effectively the the average rate for the second group (before a full moon) minus the average rate for the third group (during a full moon). When the days switch from during a full moon to before a full moon, the estimated average admission rate decreases by 2.5 patients per day.

Parameter estimates, 2 of 3

11.458 - 10.917 = 0.541
13.417 - 10.917 = 2.5

10.917 - 11.458 = -0.541
13.417 - 11.458 = 1.959
\(\ \)
Reference category, the category associated with the indicator variable left out of the model.

You have a third choice: leaving out the first indicator. The intercept now represents the estimated average for the first group (before a full moon) which is about 11.5 patients per day. The first slope tells you that you see about half a patient less on average after a full moon compared to before a full moon. The second slope tells you that you see about 2 patients more on average during a full moon compared to before a full moon.

You may see the term “reference category”. This represents the category associated with the left out indicator variable. The other categories are compared to the reference category.

Which is the best interpretation? It depends a lot on the context of the problem. I would argue that comparing before and after to during makes the most sense, but you could argue for any of these three interpretations.

Or you could synthesize these. Average admission rates are 11.5 patients per day before a full moon, increase by about 2.5 patients per day on average during a full moon and drop by about 2 patients per day after a full moon. Then admission rates jump back up by about half a patient per day when you switch back to days before a full moon.

Using moon as a fixed factor

Removing the unneeded rows

Parameter estimates using Moon as a fixed factor

Live demo, Multiple factor analysis of variance

Break #1

What you have learned
- Indicator variables for three or more categories
What’s coming next
- Multiple factor analysis of variance

Mathematical model

\(Y_{ijk} = \mu + \alpha_i + \beta_j +\epsilon_{ijk}\)
- i=1,…,a levels of the first categorical variable
- j=1,…,b levels of the second categorical variable
- k=1,…,n replicates with first and second categories

\(\ \)

\(H_0:\ \alpha_i=0\) for all i

\(\ \)

\(H_0:\ \beta_j=0\) for all j

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Live demo, Multiple factor analysis of variance

Break #2

What you have learned
- Multiple factor analysis of variance
What’s coming next
- Checking assumptions of analysis of variance

Assumptions

Normality
Equal variances
Independence

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Live demo, Checking assumptions of analysis of variance

Break #3

What you have learned
- Checking assumptions of analysis of variance
What’s coming next
- Interactions in analysis of variance

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2

3

4

Live demo, Interactions in analysis of variance

Break #4

What you have learned
- Interactions in analysis of variance
What’s coming next
- Interactions in analysis of covariance

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8

Live demo, Interactions in analysis of covariance

Break #5

What you have learned
- Interactions in analysis of covariance
What’s coming next
- Interactions in multiple linear regression

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10

Live demo, Interactions in multiple linear regression

Break #6

What you have learned
- Interactions in multiple linear regression
What’s coming next
- The general linear model

Slide 04-07

Live demo, The general linear model

Break #7

What you have learned
- The general linear model
What’s coming next
- To be determined

Slide 04-08

Live demo, To be determined

Summary

What you have learned
- Indicator variables for three or more categories
- Multiple factor analysis of variance
- Checking assumptions of analysis of variance
- Interactions in analysis of variance
- Interactions in analysis of covariance
- Interactions in multiple linear regression
- The general linear model
- To be determined

Additional topics??

Learning objectives for multifactor ANOVA
- Define Factor
- Define Level
- Assess types of ANOVA based on factors and levels
- Calculate the maximum number of groups in a multi-factorial ANOVA
- Define interaction
- Explain how to determine if an interaction between the factors exists
- Explain how to “name” a multi-factorial ANOVA
- Write the Null and Alternative Hypothesis for a multi-factorial ANOVA
- Write the Research Questions for a multi-factorial ANOVA
- Write the Research Interaction Questions for a multi-factorial ANOVA
- Define and explain the order of interpretation for a multi-factorial ANOVA
- Define Main Effects
- Explain the test of the Main Effects
- Define Simple Main Effects
- Explain the test of the Simple Main Effects
- Define Interaction Effect
- Explain the test of the Interaction Effect
- Explain what information the Interaction Effect test provides
- Explain the difference in information obtained between a 2-way ANOVA and 2 one way ANOVA’s
- Define the “Differences Between the Differences”
- Calculate the “Differences Between the Differences”
- Draw conclusions about the “Differences Between the Differences”
- Interpret the Interaction Effect test
- Determine the next steps of interpretation with a significant interaction effect
- Determine the next steps of interpretation with a non-significant interaction effect
- Interpret the Main Effects
- Interpret the Simple Main Effects
- List and define the Assumptions and tests for a multi-factorial ANOVA
- Define Marginal Means
- Explain the difference between Marginal Means and Means
- Complete a statistical write-up of a multi-factorial ANOVA with all elements included